Preference-based Search and Multi-criteria Optimization

Transcription

1 Fro: AAAI-02 Proceedings. Copyright 2002, AAAI ( All rights reserved. Preference-based Search and Multi-criteria Optiization Ulrich Junker ILOG 1681, route des Dolines F Valbonne Abstract Many real-world AI probles (e.g. in configuration) are weakly constrained, thus requiring a echanis for characterizing and finding the preferred solutions. Preferencebased search (PBS) exploits preferences between decisions to focus search to preferred solutions, but does not efficiently treat preferences on defined criteria such as the total price or quality of a configuration. We generalize PBS to copute balanced, extree, and Pareto-optial solutions for general CSP s, thus handling preferences on and between ultiple criteria. A aster-pbs selects criteria based on trade-offs and preferences and passes the as optiization objective to a sub-pbs that perfors a constraint-based Branch-and-Bound search. We project the preferences of the selected criterion to the search decisions to provide a search heuristics and to reduce search effort, thus giving the criterion a high ipact on the search. The resulting ethod will particularly be effective for CSP s with large doains that arise if configuration catalogs are large. Keywords: preferences, nononotonic reasoning, constraint satisfaction, ulti-criteria optiization, search. Introduction In this paper, we consider cobinatorial probles that are weakly constrained and that lack a clear global optiization objective. Many real-world AI probles have these characteristics: exaples can be found in configuration, design, diagnosis, but also in teporal reasoning and scheduling. An exaple for configuration is a vacation adviser syste that chooses vacation destinations fro a potentially very large catalog. Given user requireents (e.g. about desired vacation activities such as wind-surfing, canyoning), copatibility constraints between different destinations, and global resource constraints (e.g. on price) usually still leave a large set of possible solutions. In spite of this, ost of the solutions will be discarded as long as ore interesting solutions are possible. Preferences on different choices and criteria are an adequate way to characterize the interesting solutions. For exaple, the user ay prefer Hawaii to Florida for doing wind-surfing or prefer cheaper vacations in general. Different ethods for representing and treating preferences have been developed in different disciplines. In AI, Copyright c 2002, Aerican Association for Artificial Intelligence ( All rights reserved. preferences are often treated in a qualitative way and specify an order between hypotheses, default rules, or decisions. Exaples for this have been elaborated in nononotonic reasoning (Brewka 1989) and constraint satisfaction (Junker 2000). Here, preferences can be represented by a predicate or a constraint, which allows coplex preference stateents (e.g. dynaic preferences, soft preferences, etapreferences and so on). Furtherore, preferences between search decisions also allow to express search heuristics and to reduce search effort for certain kinds of scheduling probles (Junker 2000). In our vacation adviser exaple, the basic decisions consist in choosing one (or several) destinations and we can thus express preferences between individual destinations. However, the user preferences are usually forulated on global criteria such as the total price, quality, and distance which are defined in ters of the prices, qualities, and distances of all the chosen destinations. We thus obtain a ulti-criteria optiization proble. We could try to apply the preference-based search (Junker 2000) by choosing the values of the different criteria before choosing the destinations. However, this ethod has severe draw-backs: 1. Choosing the value of a defined criterion highly constrains the reaining search proble and usually leads to a thrashing behaviour. 2. The different criteria are iniized in a strict order. We get solutions that are optial w.r.t. soe lexicographic order, but none that represents coproises between the different criteria. E.g., the syste ay propose a cheap vacation of bad quality and an expensive vacation of good quality, but no coproise between price and quality. Hence, a naive application of preferences between decisions to ulti-criteria optiization probles can lead to thrashing and lacks a balancing echanis. Multi-criteria optiization avoids those proble. In Operations Research, a ulti-criteria optiization proble is usually apped to a single or a sequence of single-criterion optiization probles which are solved by traditional ethods. Furtherore, there are several notions of optiality such as Pareto-optiality, lexicographic optiality, and lexicographic ax-order optiality (Ehrgott 1997). We can thus deterine extree solutions where one criteria is 34 AAAI-02

2 favoured to another criteria as well as balanced solutions where the different criteria are as close together as possible and that represent coproises. This balancing requires that the different criteria are coparable, which is usually achieved by a standardization ethod. Surprisingly, the balancing is not achieved by weighted sus of the different criteria, but by a new lexicographic approach (Ehrgott 1997). In order to find a coproise between a good price and a good quality, Ehrgott first iniizes the axiu between (standardized versions of) price and quality, fixes one of the criteria (e.g. the standardized quality) at the resulting iniu, and then iniizes the other criterion (e.g. the price). In this paper, we will develop a odified version of preference-based search that solves a iniization subproble for finding the best value of a given criterion instead of trying out the different value assignents. Furtherore, we also show how to copute Pareto-optial and balanced solutions with new versions of preference-based search. Multi-criteria optiization as studied in Operations Research also has draw-backs. Qualitative preferences as elaborated in AI can help to address following issues: 1. We would like to state that certain criteria are ore iportant than other criteria without choosing a total ranking of the criteria as required by lexicographic optiality. For exaple, we would like to state a preference between a sall price and a high quality on the one hand and a sall distance on the other hand, but we still would like to get a solution where the price is iniized first and a solution where the quality is axiized first. 2. Multi-criteria optiization specifies preferences on defined criteria, but it does not translates the to preferences between search decisions. In general, it is not evident how to autoatically derive a search heuristics fro the selected optiization objective. Adequate preferences between search decisions provide such a heuristics and also allow to apply preference-based search to reduce the search effort for the subproble. In order to address the first point, we copare the different notions of optial solutions with the different notions of preferred solutions that have been elaborated in nononotonic reasoning. If no preferences between criteria are given, the Pareto-optial solutions correspond to the G-preferred solutions (Grosof 1991; Geffner & Pearl 1992; Junker 1997) and the lexicographic-optial solutions correspond to the B-preferred solutions (Brewka 1989; Junker 1997). Preferences between criteria can easily be taken into account by the latter ethods. For balanced solutions, we present a variant of Ehrgott s definition that respects preferences between criteria as well. The different versions of preference-based search will also respect these additional preferences. We thus obtain a syste where the user can express preferences on the criteria and preferences between the criteria and choose between extree solutions, balanced solutions and Pareto-optial solutions. As entioned above, the new versions of preferencebased search solve a iniization subproble when deterining the best value for a selected criterion. We would like to also use preference-based search for solving the subprobles. However, the preferences are only expressed on the criteria and not on the search decisions. It therefore is a natural idea to project the preferences on the selected criterion to the search decisions. We will introduce a general ethod for preference projection, which we then apply to usual objectives such as su, in, ax, and eleent constraints. It is iportant to note that these projected preferences will change fro one subproble to the other. The projected preferences will be used to guide the search and to reduce search effort. Depending on the projected preferences, copletely different parts of the search space ay be explored and, in particular, the first solution depends on the chosen objective. Search effort can be reduced since the projected preferences preserve Pareto-optiality. We therefore adapt the new preference-based search ethod for Paretooptial solutions for solving the subprobles. The paper is organized as follows: we first introduce different notions of optiality fro ulti-criteria optiization and then extend the to cover preferences between criteria. After this, we develop new versions of preference-based search for coputing the different kinds of preferred solutions. Finally, we introduce preference projection. Preferred Solutions We first introduce different notions of optiality fro ulticriteria optiization and then link the to definitions of preferred solutions fro nononotonic reasoning. Preferences on Criteria Throughout this paper, we consider cobinatorial probles that have the decision variables X := (x 1,...,x ), the criteria Z := (z 1,...,z n ), and the constraints C. Each decision variable x i has a doain D(x i ) fro which its values will be chosen. For exaple, x i ay represent the vacation destination in the i-th of = 3 weeks. The constraints in C have the for C(x 1,...,x ). Each constraint sybol C has an associated relation R C.Inour exaple, there ay be copatibility constraints (e.g., the destinations of two successive vacation destinations ust belong to neighboured countries) and requireents (e.g., at least one destination should allow wind-surfing and at least one should allow useu visits). Each criterion z i has a definition in for of a functional constraint z i := f i (x 1,...,x ) and a doain D(z i ). Exaples for criteria are price, quality, and distance (zone). The price is a su of eleent constraints: price := price(x i ) i=1 The total quality is defined as iniu of the individual qualities and the total distance is the axiu of the individual distances. The individual prices, qualities, and destinations are given by tables such as the catalog in figure 1. A solution S of (C, X ) is a set of assignents {x 1 = v 1,...,x = v } of values fro D(x i ) to each x i such that all constraints in C are satisfied, i.e. (v 1,...,v ) R C for each constraint C(x 1,...,x ) C.Wewrite v S (z i ) for the value f i (v 1,...,v ) of z i in the solution S. AAAI-02 35

3 Destination Price Quality Distance Activities Athens useus, wind-surfing Barcelona useus, wind-surfing Florence useus London useus Munich useus Nice wind-surfing... z 2 S 1 S2 S 3 S4 S 5 S 6 S 7 S 8 z 2 S 1 S 9 S 15 S2 S 10 z 3 =10 S 16 S S 3 11 S 17 z S4 3 =20 S 12 z 3 =30 S S 13 5 S 14 S 6 S 7 S 8 Figure 1: Catalog of a fictive hotel chain Furtherore, we introduce preferences between the different values for a criterion z i and thus specify a ulticriteria optiization proble. Let zi D(z i ) D(z i ) be a strict partial order for each z i.for exaple, we choose < for price and distance and > for quality. We write u v iff u v or u = v. Multiple criteria optiization provides different notions of optiality. The ost well-known exaples are Pareto optiality, lexicographic optiality, and optiality w.r.t. weighted sus. APareto-optial solution S is locally optial. If another solution S is better than S w.r.t. a criterion z j then S is better than S for soe other criterion z k : Definition 1 A solution S of (C, X ) is a Pareto-optial solution of (C, X, Z, zi ) iff there is no other solution S of (C, X ) s.t. v S (z k ) zk v S (z k ) for a k and v S (z i ) zi v S (z i ) for all i. Pareto-optial solutions narrow down the solution space since non-pareto-optial solution do not appear to be acceptable. However, their nuber is usually too large in order to enuerate the all. Figure 2 (left) shows the Paretooptial solutions S 1 to S 8 for the two criteria z 1 and z 2. Fro now on, we suppose that all the zi s are total orders. This siplifies the presentation of definitions and algoriths. A lexicographic solution requires to choose a ranking of the different criteria. We express it by a perutation of the indices: Definition 2 Let π be a perutation of 1,...,n. Let V S (π(z)) := (v S (z π1 ),...,v S (z πn )). A solution S of (C, X ) is an extree solution of (C, X, Z, zi ) iff there is no other solution S of (C, X ) s.t. V S (π(z)) lex V S (π(z)). Different rankings lead to different extree 1 solutions which are all Pareto-optial. In figure 2 (left), we obtain the extree solutions S 1 where z 1 is preferred to z 2 and S 8 where z 2 is preferred to z 1. Extree solutions can be deterined by solving a sequence of single-criterion optiization probles starting with the ost iportant criterion. If we cannot establish a preference order between different criteria then we would like to be able to find coproises between the. Although weighted sus (with equal weights) are often used to achieve those coproises, they do not necessarily produce the ost balanced solutions. If 1 We use the ter extree in the sense that certain criteria have an absolute priority over other criteria. z 1 Figure 2: Pareto-optial solutions for iniization criteria we choose the sae weights for z 1 and z 2,weobtain S 7 as the optial solution. Furtherore, if we slightly increase the weight of z 1 the optial solution jups fro S 7 to S 2. Hence, weighted sus, despite of their frequent use, do not appear a good ethod for balancing. In (Ehrgott 1997), Ehrgott uses lexicographic axorderings to deterine optial solutions. In this approach, values of different criteria need to be coparable. For this purpose, we assue that the preference orders zi of the different criteria are equal to a fixed order D. This usually requires soe scaling or standardization of the different criteria. We also introduce the reverse order D which satisfies z i D z j iff z j D z i. When coparing two solutions S 1 and S 2, the values of the criteria in each solution are first sorted w.r.t. the order D before being copared by a lexicographic order. This can lead to different perutations of the criteria for different solutions. We describe the sorting by a perutation ρ S that depends on a given solution S and that satisfies two conditions: 1. ρ S sorts the criteria in a decreasing order: if v S (z i ) D v S (z j ) then ρ S i <ρ S j. 2. ρ S does not change the order if two criteria have the sae value: if i<jand v S (z i )=v S (z j ) then ρ S i <ρ S j. Definition 3 A solution S of (C, X ) is a balanced solution of (C, X, Z, D ) iff there is no other solution S of (C, X ) s.t. V S (ρ S (Z)) lex V S (ρ S (Z)). Balanced solutions are Pareto-optial and they are those Pareto-optial solutions where the different criteria are as close together as possible. In the exaple of figure 2 (left), we obtain S 5 as balanced solution. According to Ehrgott, it can be deterined as follows: first ax(z 1,z 2 ) is iniized. If is the resulting optiu, the constraint ax(z 1,z 2 )= is added before in(z 1,z 2 ) is iniized. Balanced solutions can thus be deterined by solving a sequence of single-criterion optiization probles. Preferences between Criteria If any criteria are given it is natural to specify preferences between different criteria as well. For exaple, we would like to specify that a (sall) price is ore iportant than a (short) distance without specifying anything about the quality. We therefore introduce preferences between criteria in for of a strict partial order Z Z Z. z 1 36 AAAI-02

4 Preferences on criteria and between criteria can be aggregated to preferences between assignents of the for z i = v. Let be the sallest relation satisfying following two conditions: 1. If u zi v then (z i = u) (z i = v) for all u, v and 2. If z i Z z j then (z i = u) (z j = v) for all u, v. Hence, if a criteria z i is ore iportant than z j then any assignent to z i is ore iportant than any assignent to z j. In general, we could also have preferences between individual value assignents of different criteria. In this paper, we siplified the structure of the preferences in order to keep the presentation siple. In nononotonic reasoning, those preferences between assignents can be used in two different ways: 1. as specification of a preference order between solutions. 2. as (incoplete) specification of a total order (or ranking) between all assignents, which is in turn used to define a lexicographic order between solutions. The ceteris-paribus preferences (Boutilier et al. 1997) and the G-preferred solutions of (Grosof 1991; Geffner & Pearl 1992) follow the first approach, whereas the second approach leads to the B-preferred solutions of (Brewka 1989; Junker 1997). We adapt the definitions in (Junker 1997) to the specific preference structure of this paper: Definition 4 A solution S of (C, X ) is a G-preferred solution of (C, X, Z, ) if there is no other solution S of (C, X ) such that v S (z k ) v S (z k ) for soe k and for all i with v S (z i ) zi v S (z i ) there exists a j s.t. z j Z z i and v S (z j ) zj v S (z j ). Hence, a criterion can becoe worse if a ore iportant criterion is iproved. In figure 2 (right), S 1 to S 8 are G- preferred if z 1 Z z 3 and z 2 Z z 3 are given. Each G- preferred solution corresponds to a Pareto-optial solution. If there are no preferences between criteria, each Paretooptial solution corresponds to soe G-preferred solution. However, if there are preferences between criteria, certain Pareto-optial solutions S are not G-preferred. There can be a G-preferred solution S that is better than S for a criterion z i,but worse for a less iportant criterion z j (i.e. z i Z z j ). In general, we ay get new G-preferred solutions if we add new constraints to our proble. However, adding upper bounds on criteria does not add new G-preferred solutions: Proposition 1 S is a G-preferred solution of (C {z i zi u}, X, Z, ) iff S is a G-preferred solution of (C, X, Z, ) and v S (z i ) zi u. Although this property appears to be trivial it is not satisfied for the B-preferred solutions which will be introduced next. It will be essential for coputing G-preferred solutions. In the definition of lexicographic optial solutions, a single ranking of the given criteria is considered. In the definition of B-preferred solutions, we consider all rankings that respect the given preferences between the criteria. Following definition has been adapted fro (Brewka 1989; Junker 1997) to our specific preference structure: Definition 5 A solution S of (C, X ) is a B-preferred solution of (C, X, Z, ) if there exists a perutation π such that 1. π respects Z (i.e. z i Z z j iplies π i < π j ) and 2. there is no other solution S of (C, X ) satisfying V S (π(z)) lex V S (π(z)). The B-preferred solution for π can be coputed by solving a sequence of iniization probles: Let A 0 := and A i := A i 1 {z πi = } where = in zπi {v C A i 1 {z πi = v} has a solution} In figure 2 (right), S 1 and S 8 are B-preferred (for z 1 Z z 3 and z 2 Z z 3 ). Each B-preferred solution corresponds to an extree solution. If there are no preferences between criteria, each extree solution corresponds to soe B-preferred solution. If there are preferences between criteria certain extree solutions ay not be B-preferred. For exaple, S 15 is an extree solution, which is obtained if first the distance is iniized and then the price. However, this ranking of criteria does not respect the given preferences. In (Junker 1997), it has been shown that each B-preferred solution is a G-preferred one, but that the converse is not true in general. In figure 2 (right), S 2 to S 6 are G-preferred, but not B-preferred. These solutions assign a worse value to z 1 than the B-preferred solution S 1,but a better value than S 8. Siilarly, they assign a better value to z 2 than S 8,but aworse value than S 1.Itisevident that such a case cannot arise if each criteria has only two possible values. Hence, we get an equivalence in following case, where no coproises are possible: Proposition 2 If there is no z i such that v 1 zi v 2 zi v 3 and C {z = v i } has a solution for i =1, 2, 3 then each G-preferred solution of (C, X, Z, ) is also a B-preferred solution of (C, X, Z, ). So far, we siply adapted existing notions of preferred solutions to our preference structure and related the to well-known notions of optiality. We now introduce a new kind of preferred solutions that generalizes the balanced solutions. We want to be able to balance certain criteria, e.g. the price and the quality, but prefer these two criteria to other criteria such as the distance. Hence, we liit the balancing to certain groups of criteria instead of finding a coproise between all criteria. For this purpose, we partition Z into disjoint sets G 1,...,G k of criteria. Given a criterion z, we also denote its group by G(z). The criteria in a single group G i will be balanced. The groups theselves are handled by using a lexicographic approach. Thus, we can treat preferences between different groups, but not between different criteria of a single group. Given a strict partial order G between the G i s, wecan easily define an order Z between criteria: if G 1 G 2 and z i G 1, z j G 2 then z i Z z j. We now cobine definitions 5 and 3. As in definition 5, we first choose a global perutation π that respects the preferences between groups. We then locally sort the values of each balancing group in a decreasing order. We describe this local sorting by a perutation θ S that depends on a given solution S and that satisfies three conditions: 1. θ S can only exchange variables that belong to the sae balanced group: G(z i )=G(z θ S i ). AAAI-02 37

5 Algorith Extree-PBS1(C, Z, ) 1. A := ; U := Z; Q := ; 2. while Q U do 3. for all (z = q) Q do 4. let be iniize(c A, z, z) 5. if q z then Q := Q {z = q}; 6. if U = and Q then fail; 7. B := {y U y U : y Z y and 8. (y = q) Q}; 9. if B = then fail else 10. select z B; 11. let be iniize(c A, z, z); 12. choose A := A {z = }; U := U {z} 13. or Q := Q {z = }; 14. return A; Figure 3: Algorith Extree-PBS1 distance=4 quality=1 1st solution price=160 distance=4 quality=1 distance=2 price=250 2nd solution price=160 quality=4 distance=2 price=250 Figure 4: Finding extree solutions quality=4 2. θ S sorts the criteria of each group in a decreasing order: if v S (z i ) D v S (z j ) and G(z i )=G(z j ) then θi S <θj S. 3. θ S does not change the order if two criteria of the sae group have the sae value: if i<j, v S (z i )=v S (z j ), and G(z i )=G(z j ) then θi S <θj S. Definition 6 A solution S of (C, X ) is an E-preferred solution of (C, X, Z, ) if there exists a perutation π such that 1. π respects Z (i.e. z i Z z j iplies π i < π j ) and 2. there is no other solution S of (C, X ) s.t. V S (θ S (π(z))) lex V S (θ S (π(z))). In figure 2 (right), S 5 is E-preferred (z 1 Z z 3 and z 2 Z z 3 ). Each E-preferred solution corresponds to a balanced solution. If there are no preferences between criteria, each balanced solution corresponds to an E-preferred solution. Interestingly, we can ap E-preferred solutions to B- preferred solutions if we introduce suitable variables and preferences. For each group G of cardinality n G,weuse following in-ax-variables y G,nG,...,y G,1 : y G,i := in{ax(x) X G s.t. X = i} (1) where ax(x) := ax{z z X}. Let Ẑ be the set of all of these in-ax-variables. Following preferences ensure that in-ax-variables for larger subsets X are ore iportant: y G,i ˆ y G,i 1 for i = n G,...,2 (2) A preference between a group G and a group G can be translated into a preference between the last in-axvariable of G and the first one of G: y G,1 ˆ y G,nG (3) The E-preferred solutions then correspond to the B-preferred solutions of the translated criteria and preferences: Theore 1 S is an E-preferred solution of (C, X, Z, ) iff S is a B-preferred solution of (C, X, Ẑ, ˆ ). We have thus established variants of Pareto-optial, extree, and balanced solutions that take into account preferences between criteria. On the one hand, we gain a better understanding of the existing preferred solutions by this coparison with notions for ulti-criteria optiization. On the other hand, we obtain a balancing echanis that fits well into the qualitative preference fraework. Preference-based Search We now adapt the preference-based search algorith fro (Junker 2000) to treat preferences on criteria and to copute Pareto-optial solutions and balanced solutions as well. Extree Solutions The algorith in (Junker 2000) can easily be adapted to preferences on criteria. The resulting algorith is shown in figure 3. We explain its basic idea for the sall exaple shown in figure 4, where price and quality are preferred to distance. The algorith aintains a set U of unexplored criteria, which is initialized with the set of all criteria (i.e. price, quality, and distance). In each step, the algorith selects a best criterion z of U (e.g. the price). Instead of trying to assign different values to the total price, we deterine the cheapest price by solving a iniization subproble: iniize(a, z, z ):= (4) in z {v A {z = v} has a solution} In order to obtain a unique result, we assue that the orders zi are strict total orders throughout the entire section. In our exaple, the cheapest solution has a price of 160. We now add the assignent price = 160 to an initially epty set A of assignents. In figure 3, the eleents of A occur as labels of the left branches. We then deterine the best quality under this assignent. Once the price and quality have been deterined we can deterine a distance as well. In order to find further solutions, Extree-PBS1 does not add the negation of assignents, but introduces a refutation query for each assignent z = v. Wesay that z = v is d if it becoes inconsistent after assigning values to unexplored criteria that ay precede z. The refutation queries are added to a set Q. Wecan reove an eleent fro Q if it has been d by further assignents. The assignent to the distance cannot be d since there are no further unexplored criteria. The quality of 1 cannot be d since the single non-explored criterion distance cannot precede the 38 AAAI-02

6 quality. However, we can the price of 160 by first axiizing the quality. After this, we can again iniize the price and the distance, which leads to a new solution as shown in figure 3. Theore 2 Algorith Extree-PBS1(C, Z, ) always terinates. Each successful run returns a B-preferred solution of (C, X, Z, ) and each such B-preferred solution is returned by exactly one successful run. According to theore 1, we can use this algorith to also copute balanced solutions supposed we provide it with the adequately translated criteria and preferences. Pareto-optial Solutions The algorith for B-preferred solutions is thus relatively siple. Coputing G-preferred solutions turns out to be ore subtle. Interestingly, ost operations of algorith Extree-PBS1 are also valid for G-preferred solutions except for the rules in lines 6 and 8, where the algorith backtracks since no B-preferred solution exists that is copatible with the given assignents and refutation queries. An algorith for G-preferred solutions cannot backtrack in this case since there ay be G-preferred solutions, which are not B- preferred. In order to obtain an algorith for coputing G- preferred solutions, we have to avoid such a situation. The basic idea is to add additional constraints that produce the equivalence between G-preferred and B-preferred solutions that is stated in proposition 2. We need to reduce the doain of each criterion z such that all values are either a best or a worst eleent of the doain. If there are interediate values u between a best value q and a worst value v then we consider two possibilities: either we ipose u as upper bound on z by adding a constraint z z u or we require that z = u is d. In general, adding constraints can introduce new G-preferred solutions. Proposition 1 states that this is not the case if upper bounds are added. The resulting algorith is given in figure 5. It needs the set of values that have not yet been eliinated by soe upper bound: Pos(A, z) :={v (z u) A : u z v} (5) The algorith 5 deterines all G-preferred solutions: Theore 3 Algorith Pareto-PBS1(C, Z, ) always terinates. Each successful run returns a G-preferred solution of (C, X, Z, ) and each G-preferred solution is returned by exactly one successful run. Adding upper bounds thus helps to control the behaviour of PBS and to avoid that it jups fro one extree solution to the other. In figure 4, iniizing the quality s the cheapest price of 160, asitwas required. However, we now obtain a very high price of 250. Inorder to obtain coproises between price and quality, we constrain the price to be strictly saller than 250 before iniizing the quality. This discussion indicates that we need not consider all upper bounds. We first deterine an extree solution that leads to a high price and we use this value to ipose an upper bound on the price. Hence, we can extract the values of the upper bounds fro the previous solutions found so far. Algorith Pareto-PBS1(C, Z, ) 1. A := ; U := Z; Q := ; 2. while Q U do 3. for all (z = q) Q do 4. let be iniize(c A, z, z) 5. if q z then Q := Q {z = q}; 6. if U = and Q then fail; 7. B := {y U y U : y Z y and 8. (y = q) Q}; 9. if there is (z = q) Q and v, w Pos(A, z) 10. s.t. q z v z w then 11. select a z-inial eleent u 12. s.t. u Pos(A, z) and q z u z w; 13. choose A := A {z z u}; 14. or Q := Q {z = u}; 15. else if B = then fail else 16. select z B; 17. let be iniize(c A, z, z); 18. A := A {z z }; 19. choose A := A {z = }; U := U {z} 20. or Q := Q {z = }; 21. return A; Figure 5: Algorith Pareto-PBS1 Preference Projection A ulti-criteria optiization proble is often solved by a sequence of single-criterion optiization probles having different objectives. We can, for exaple, solve each of these subprobles by a constraint-based Branch-and-Bound which aintains the best objective value found so far. Now, when changing the objective, the search heuristics should be adapted as well. It is a natural idea to project the preference order of the objective to the decision variables that appear in its definition. We define preference projection as follows: Definition 7 xk is a projection of zj via f j (x 1,...,x ) to x k if and only if following condition holds for all u 1,...,u and v 1,...,v with u i = v i for i =1,...,k 1,k+1,...,: if u k xk v k then f j (u 1,...,u ) y f j (v 1,...,v ) Definition 8 x1,..., x is a projection of z1,..., zn via f 1,...,f n to x 1,...,x if xi is a projection of zj via f j (x 1,...,x ) to x i for all i, j. The projected preferences preserve Pareto-optiality: Theore 4 Let x1,..., x be a projection of z1,..., zn via f 1,...,f n to x 1,...,x. If S is a Paretooptial solution w.r.t. the criteria z 1,...,z n and the preferences z1,..., zn then there exists a solution S that 1. is apareto-optial solution w.r.t. the criteria x 1,...,x and the preferences x1,..., x and 2. v S (z i )=v S (z i ) for all criteria z i. Since extree and balanced solutions are Pareto-optial we can additionally use the projected preferences to reduce search effort when solving a sub-proble. For this purpose, we adapt the algorith Pareto-PBS1 to the decision (6) AAAI-02 39

7 variables x 1,...,x and do consistency checks instead of iniize-calls. We give soe exaples for preference projections: 1. The increasing order < is a projection of < via su, in, ax, and ultiplication with a positive coefficient. 2. The decreasing order > is a projection of < via a ultiplication with a negative coefficient. 3. Given an eleent constraint of the for y = f(x) that aps each possible value i of x to a value f(i), the following order x is a projection of < to x via f(x): u x v iff f(u) <f(v) (7) In our vacation exaple, the price, quality, and distance of a destination are all defined by eleent constraints. If we change the objective, we project preferences over a different eleent constraint. Since the projected preferences depend on the values f (e.g. price, quality), changing the objective will copletely change the order of the value assignents x i = u. Thus, the objective will have a strong ipact on the search heuristics. On the one hand, the first solution detected is influenced by the projected preferences and thus ay change when the objective is changed. On the other hand, search effort is reduced depending on the objective. Conclusion Although Preference-based Search (Junker 2000) provided an interesting technique for reducing search effort based on preferences, it could only take into account preferences between search decisions, was liited to cobinatorial probles of a special structure, and did not provide any ethod for finding coproises in absence of preferences. In this paper, we have lifted PBS fro preferences on decisions to preferences on criteria as they are coon in qualitative decision theory (Doyle & Thoason 1999; Bacchus & Grove 1995; Boutilier et al. 1997; Doshlak, Brafan, & Shiony 2001). We further generalized PBS such that not only extree solutions are coputed, but also balanced and Pareto-optial solutions. Surprisingly, balanced solutions can be coputed by a odified lexicographic approach (Ehrgott 1997) which fits well into a qualitative preference fraework as studied in nononotonic reasoning and qualitative decision theory. Our search procedure consists of two odules. A aster- PBS explores the criteria in different orders and assigns optial values to the. The optial value of a selected criterion is deterined by a sub-pbs, which perfors a constraintbased Branch-and-Bound search through the original proble space (i.e. the different value assignents to decision variables). We furtherore project the preferences on the selected criterion to preferences between the search decisions, which provides an adapted search heuristics for the optiization objective and which allows to reduce search effort further. Hence, different regions of the search space will be explored depending on the selected objective. Our approach has been ipleented in ILOG JCONFIGURATOR V2.0 and adds ulti-criteria optiization functionalities to this constraint-based configuration tool. Other CSP-based approaches to ulti-criteria optiization are doing a single Branch-and-Bound search for all criteria, which requires to aintain a set of non-doinated solutions (cf. (Boutilier et al. 1997), (Gavanelli 2002)) instead of a single bound. Doinance checking ensures that nonpreferred solutions are pruned. Interestingly, the Master- PBS does not need doinance checking, but uses refutation queries to avoid non-preferred solutions. Future work will be devoted to iprove the pruning behaviour of the new PBS procedures by incorporating the conflict checking ethods of (Junker 2000). We will also exaine whether PBS can be used to deterine preferred solutions as defined by soft constraints (Khatib et al. 2001). Acknowledgeents For helpful coents and discussions, I would like to thank Olivier Lhoe, Xavier Ceugniet, Daniel Mailharro, Mark Wallace, as well as the anonyous reviewers. References Bacchus, F., and Grove, A Graphical odels for preference and utility. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, Boutilier, C.; Brafan, R.; Geib, C.; and Poole, D A constraint-based approach to preference elicitation and decision aking. In Doyle, J., and Thoason, R. H., eds., Working Papers of the AAAI Spring Syposiu on Qualitative Preferences in Deliberation and Practical Reasoning. Brewka, G Preferred subtheories: An extended logical fraework for default reasoning. In IJCAI-89, Doshlak, C.; Brafan, R.; and Shiony, E Preference-based configuration of web-page content. In IJCAI Doyle, J., and Thoason, R. H Background to qualitative decision theory. AI Magazine 20(2): Ehrgott, M A characterization of lexicographic ax-ordering solutions. In Methods of Multicriteria Decision Theory: Proceedings of the 6th Workshop of the DGOR Working-Group Multicriteria Optiization and Decision Theory, Egelsbach: Häsel-Hohenhausen. Gavanelli, M An ipleentation of Pareto optiality in CLP(FD). In CP-AI-OR 02. Geffner, H., and Pearl, J Conditional entailent: Bridging two approaches to default reasoning. Artificial Intelligence 53: Grosof, B Generalizing prioritization. In KR 91, Cabridge, MA: Morgan Kaufann. Junker, U A cuulative-odel seantics for dynaic preferences on assuptions. In IJCAI-97, Junker, U Preference-based search for scheduling. In AAAI-2000, Khatib, L.; Morris, P.; Morris, R. A.; and Rossi, F Teporal constraint reasoning with preferences. In IJCAI, AAAI-02